I think checking back the flop is fine. He's definitely not folding any pair/Ax and might even station just some random overcards + backdoor straight and whatnot. If he checks turn again we can just bet a lot of turns instead.

As played turn seems like a call. Raising seems a little bit thin, and we want to keep his bluffs in since we're calling on almost all rivers!

Once he checks river I would just check back though. There is not that much to get value from, and he can definitely play some Ax like this. So just check down and take your showdown.